The ground state success probability of quantum annealing is usually derived under the unique ground state condition in an asymptotic sense, that is, we obtain some expressions or bounds for the leading terms of the ground state success probability by taking \(t_f\)  to go to infinity [1]}, [2]}, [3]}, [4]}.
We provide the probability lower bound in (REF ) that is for finite \(t_f\)  and does not impose the unique ground state restriction.
From the proof, \(p_0\)  can be interpreted as the probability that the quantum system stays in a ground state during the annealing procedure.
We note that
\(\frac{d {\mathbf {H}}_{QA}(u\,t_f)}{d u} = \frac{d A(u \,t_f)}{d u} \, {\mathbf {H}}_0 + \frac{d B(u \,t_f)}{d u} \, {\mathbf {H}}_1,\) 
